Sharp Favard Length of Random Cantor Sets

Abstract

We show that for a large class of planar 1-dimensional random fractals $S$ S , the Favard length $\operatorname{Fav}(S(r))$ Fav ( S ( r ) ) of the neighborhood $S(r)$ S ( r ) is comparable to $\log ^{-1}(1/r)$ log − 1 ( 1 / r ) , matching a universal lower bound; up to now, this was only known in expectation for a few concrete models. In particular, we show that there exist 1-Ahlfors regular sets with the fastest possible Favard length decay. For a wide class of planar one-dimensional “grid random fractals”, including fractal percolation and its Ahlfors-regular variants, we further show that $\operatorname{Fav}(S(r))/\log (1/r)$ Fav ( S ( r ) ) / log ( 1 / r ) converges almost surely, and we identify the limit explicitly. Furthermore, we prove that for some 1-dimensional Ahlfors-regular random fractals $S$ S , the Favard length of $S(r)$ S ( r ) decays instead like $\log \log (1/r)/\log (1/r)$ log log ( 1 / r ) / log ( 1 / r ) , showing that the $1/\log (1/r)$ 1 / log ( 1 / r ) decay is not universal among random fractals, as might be expected from previous results.